Threshold Corrections To DY and Higgs at N 3 LO QCD

Transcription

1 Threshold Corrections To DY and Higgs at N 3 LO QCD Taushif Ahmed Institute of Mathematical Sciences, India July 2, 2015 Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 1

2 Prologue : Processes of Interest Our processes of interest Drell-Yan (DY) pp γ /Z l + l : Xsection and rapidity distributions. Higgs boson production through gg : Rapidity distributions. Higgs boson production through b b : Xsection and rapidity distributions. Such processes are very important for precision studies in SM. Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 2

3 Prologue : State of the Art in Early 2014 DY Xsection : NNLO QCD Hamberg, Matsuura, Van Neerven ( 91) N 3 LO psv Moch, Vogt (2005); Ravindran ( 06) Higgs in gg Rapidity: NNLO QCD Anastasiou, Dixon, Melnikov, Petriello ( 03); Melnikov, Petriello ( 06) N 3 LO psv Ravindran, Smith, van Neerven ( 07) Xsection : NNLO QCD Harlander, Kilgore ( 02); Anastasiou, Melnikov ( 02); Ravindran, Smith, van Neerven ( 03) N 3 LO psv Moch, Vogt ( 05); Ravindran ( 06) Higgs in b b Rapidity : NNLO QCD Anastasiou, Melnikov, Petriello ( 04, 05) N 3 LO psv Ravindran, Smith, van Neerven ( 07) Xsection : NNLO QCD Harlander, Kilgore ( 03) N 3 LO psv Ravindran ( 06) Rapidity : NNLO QCD Buehler, Herzog, Lazopoulos, Mueller ( 12) N 3 LO psv Ravindran, Smith, van Neerven ( 07) Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 3

4 Prologue : What next? Can we push the theoretical boundary little more? Testing SM with more accuracy to answer many unanswered questions Reducing the unphysical scale uncertainties First crucial step towards this direction Higgs production through gg fusion at threshold N 3 LO by Anastasiou, Duhr, Dulat, Furlan, Gehrmann, Herzog, Mistlberger ( 14) Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 4

5 Prologue : What next? Can we push the theoretical boundary little more? Testing SM with more accuracy to answer many unanswered questions Reducing the unphysical scale uncertainties First crucial step towards this direction Higgs production through gg fusion at threshold N 3 LO by Anastasiou, Duhr, Dulat, Furlan, Gehrmann, Herzog, Mistlberger ( 14) Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 4

6 Prologue : Spin-offs from Higgs Production at Threshold N 3 LO Recent results on threshold N 3 LO QCD corrections TA, Mahakhud, Mandal, Rana, Ravindran dσ/dq 2 of DY Phys. Rev. Lett. 113 (2014) dσ/dy of Higgs in gg fusion and d 2 σ/dq 2 /dy of DY Phys. Rev. Lett. 113 (2014) σ tot and dσ/dy of Higgs in b b annihilation JHEP 1410 (2014) 139 JHEP 1502 (2015) 131 Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 5

7 Prologue : Spin-offs from Higgs Production at Threshold N 3 LO Recent results on threshold N 3 LO QCD corrections TA, Mahakhud, Mandal, Rana, Ravindran dσ/dq 2 of DY Phys. Rev. Lett. 113 (2014) Goal dσ/dy of Higgs in gg fusion and d 2 σ/dq 2 /dy of DY Phys. Rev. Lett. 113 (2014) σ tot and dσ/dy of Higgs in b b annihilation JHEP 1410 (2014) 139 JHEP 1502 (2015) 131 Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 6

8 Plan of the Talk What is threshold corrections? Prescription of Computation Factorization of Soft Gluons Overall Operator Renormalization Form Factor Soft-Collinear Distribution Mass Factorization Master Formula Result of DY SV Xsection at N 3 LO Conclusions Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 7

9 Why threshold corrections? Basic calculational framework for inclusive production : QCD factorization theorem σ X ( τ, q 2 ) = 1 S ab 1 τ q2 S, q2 m 2 l + l for DY and m 2 H for Higgs τ dx x Φ ab (x) ab ( τ x, q2) Perturbatively calculable partonic cross section (CS) Non-perturbative partonic flux Φ ab (x) = ab ŝ ˆσ ab 1 x du ( x ) u f a (u) f b u Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 8

10 Why threshold corrections? Basic calculational framework for inclusive production : QCD factorization theorem σ X ( τ, q 2 ) = 1 S ab 1 τ q2 S, q2 m 2 l + l for DY and m 2 H for Higgs τ dx x Φ ab (x) ab ( τ x, q2) Perturbatively calculable partonic cross section (CS) Non-perturbative partonic flux Φ ab (x) = ab ŝ ˆσ ab 1 x du ( x ) u f a (u) f b u Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 8

11 Why Threshold... Φ ab (x) becomes large when x x min = τ enhances σ X x τ ŝ q 2 partonic COM energy is just enough to produce final state particle all the radiations are soft soft limit. Dominant contributions to this region arise from Virtual and Soft gluon emission processes (SV). For Higgs and DY productions, substantial contribution come from this region. e.g. for DY 91% - 95% at NNLO Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 9

12 Threshold Expan : Soft-Plus-Virtual (SV) Splitting the partonic CS into singular & regular parts around z τ x = q2 ŝ = 1 sing (z) SV (z) = SV δ (z) = sing (z) + hard (z) δ(1 z)+ j=0 SV j D j D j ( ln j ) (1 z) 1 z + These two are the most singular terms when z 1, so dominant contribution comes from these. hard (z) : subleading and polynomial in ln(1 z). Plus distribution + is defined by its action on test function f(z) 1 1 dzd j (z)f(z) dz lnj (1 z) [f(z) f(1)] z Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 10

13 Goal Goal : Computation of SV (z) Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 11

14 Various Methods to Compute SV Matrix elements square and phase space integrals in the soft limit. Form factors and DGLAP kernels : Catani et al; Harlander and Kilgore Factorization theorem Renormalization group invariance Sudakov resummation of form factors Soft Collinear effective theory Moch and Vogt; Ravindran, Smith and Van Neerven Becher and Neubert Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 12

15 Various Methods to Compute SV Matrix elements square and phase space integrals in the soft limit. Catani et al; Harlander and Kilgore Form factors and DGLAP kernels : Factorization theorem Renormalization group invariance Sudakov resummation of form factors Soft Collinear effective theory Moch and Vogt; Ravindran, Smith and Van Neerven Becher and Neubert Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 13

16 Structure of Partonic XSection General structure of upon taking radiative corrections : (z, q 2 ) = δ(1 z) ( + a s q 2 ) [ ( ) c (1) ln (1 z) 1 δ (1 z) + c(1) 2 (1 z) + a 2 ( s q 2 ) [ + + R 2 (z, q )] 2 + z = q2 ŝ, R i(z, q 2 ) are remaining terms. Virtual δ(1 z) + ] + R 1 (z, q 2 ) Real emission δ(1 z), D j, R k Soft approximation z 1 Defn.: a s α s/4π Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 14

17 Structure of Partonic XSection General structure of upon taking radiative corrections : (z, q 2 ) = δ(1 z) ( + a s q 2 ) [ ( ) c (1) ln (1 z) 1 δ (1 z) + c(1) 2 (1 z) + a 2 ( s q 2 ) [ + + R 2 (z, q )] 2 + z = q2 ŝ, R i(z, q 2 ) are remaining terms. Virtual δ(1 z) + ] + R 1 (z, q 2 ) Real emission δ(1 z), D j, R k Soft approximation z 1 Defn.: a s α s/4π Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 14

18 Structure of Partonic XSection General structure of upon taking radiative corrections : (z, q 2 ) = δ(1 z) ( + a s q 2 ) [ ( ) c (1) ln (1 z) 1 δ (1 z) + c(1) 2 (1 z) + a 2 ( s q 2 ) [ + + R 2 (z, q )] 2 + z = q2 ŝ, R i(z, q 2 ) are remaining terms. Virtual δ(1 z) + ] + R 1 (z, q 2 ) Real emission δ(1 z), D j, R k Soft approximation z 1 Defn.: a s α s/4π Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 14

19 Structure of Partonic XSection General structure of upon taking radiative corrections : (z, q 2 ) = δ(1 z) ( + a s q 2 ) [ ( ) c (1) ln (1 z) 1 δ (1 z) + c(1) 2 (1 z) + a 2 ( s q 2 ) [ + + R 2 (z, q )] 2 + z = q2 ŝ, R i(z, q 2 ) are remaining terms. Virtual δ(1 z) + ] + R 1 (z, q 2 ) Real emission δ(1 z), D j, R k Soft approximation z 1 Defn.: a s α s/4π Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 14

20 Factorization of Soft Gluons Soft distribution factorizes as exponentiation ( z, q 2) = S ( z, q 2, µ 2 ) [ R δ (1 z) + a s R1 (z, q 2 ) + a 2 R ] s 2 (z, q 2 ) and S ( ( ) z, q 2, µ 2 ) R = C exp Φ(z, q 2, µ 2 R), Φ is a finite distribution Defn: Ce f(z) = δ(1 z) + 1 1! f(z) + 1 f(z) f(z) + 2! is Mellin convolution. Drop all regular functions. Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 15

21 Factorization of Soft Gluons Soft distribution factorizes as exponentiation ( z, q 2) = S ( z, q 2, µ 2 ) [ R δ (1 z) + a s R1 (z, q 2 ) + a 2 R ] s 2 (z, q 2 ) and S ( ( ) z, q 2, µ 2 ) R = C exp Φ(z, q 2, µ 2 R), Φ is a finite distribution Defn: Ce f(z) = δ(1 z) + 1 1! f(z) + 1 f(z) f(z) + 2! is Mellin convolution. Drop all regular functions. Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 15

22 SV Cross Section : Conjecture We propose UV renormalized ˆ SV,I ( z, q 2, µ 2 R) = Ce 2Φ I (â s,q 2,µ 2,z) ( Z I ( â s, µ 2 R, µ 2)) 2 ˆF I ( â s, Q 2, µ 2) 2 δ(1 z) I = q, g, b for DY, Higgs through gg and Higgs in b b. Soft contribution Form factor factorizes due to Born kinematics. Overall renormalization constant required for effective Lagrangian of Higgs ( Z I = 1 for DY) Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 16

23 SV Cross Section : Conjecture We propose UV renormalized ˆ SV,I ( z, q 2, µ 2 R) = Ce 2Φ I (â s,q 2,µ 2,z) ( Z I ( â s, µ 2 R, µ 2)) 2 ˆF I ( â s, Q 2, µ 2) 2 δ(1 z) I = q, g, b for DY, Higgs through gg and Higgs in b b. Soft contribution Form factor factorizes due to Born kinematics. Overall renormalization constant required for effective Lagrangian of Higgs ( Z I = 1 for DY) Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 16

24 SV Cross Section : Conjecture We propose UV renormalized ˆ SV,I ( z, q 2, µ 2 R) = Ce 2Φ I (â s,q 2,µ 2,z) ( Z I ( â s, µ 2 R, µ 2)) 2 ˆF I ( â s, Q 2, µ 2) 2 δ(1 z) I = q, g, b for DY, Higgs through gg and Higgs in b b. Soft contribution Form factor factorizes due to Born kinematics. Overall renormalization constant required for effective Lagrangian of Higgs ( Z I = 1 for DY) Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 16

25 SV Cross Section : Conjecture We propose UV renormalized ˆ SV,I ( z, q 2, µ 2 R) = Ce 2Φ I (â s,q 2,µ 2,z) ( Z I ( â s, µ 2 R, µ 2)) 2 ˆF I ( â s, Q 2, µ 2) 2 δ(1 z) I = q, g, b for DY, Higgs through gg and Higgs in b b. Soft contribution Form factor factorizes due to Born kinematics. Overall renormalization constant required for effective Lagrangian of Higgs ( Z I = 1 for DY) Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 16

26 Cancellation of Divergences ˆ SV,I ( z, q 2, µ 2 R) = Ce 2Φ I (â s,q 2,µ 2,z) ( Z I ( â s, µ 2 R, µ 2)) 2 ˆF I ( â s, Q 2, µ 2) 2 δ(1 z) â s renormalization and Z I makes partonic CS UV finite. Soft divergences from real radiation and virtual correction cancel upon summing over all the final states (KLN). Final state collinear singularities also cancel upon summing over all the final states (KLN) Still XSection is NOT fully finite due to initial state collinear singularities This can arise from virtual as well as real corrections Mass factorization to make collinear finite. Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 17

27 Cancellation of Divergences ˆ SV,I ( z, q 2, µ 2 R) = Ce 2Φ I (â s,q 2,µ 2,z) ( Z I ( â s, µ 2 R, µ 2)) 2 ˆF I ( â s, Q 2, µ 2) 2 δ(1 z) â s renormalization and Z I makes partonic CS UV finite. Soft divergences from real radiation and virtual correction cancel upon summing over all the final states (KLN). Final state collinear singularities also cancel upon summing over all the final states (KLN) Still XSection is NOT fully finite due to initial state collinear singularities This can arise from virtual as well as real corrections Mass factorization to make collinear finite. Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 17

28 Cancellation of Divergences ˆ SV,I ( z, q 2, µ 2 R) = Ce 2Φ I (â s,q 2,µ 2,z) ( Z I ( â s, µ 2 R, µ 2)) 2 ˆF I ( â s, Q 2, µ 2) 2 δ(1 z) â s renormalization and Z I makes partonic CS UV finite. Soft divergences from real radiation and virtual correction cancel upon summing over all the final states (KLN). Final state collinear singularities also cancel upon summing over all the final states (KLN) Still XSection is NOT fully finite due to initial state collinear singularities This can arise from virtual as well as real corrections Mass factorization to make collinear finite. Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 17

29 Cancellation of Divergences ˆ SV,I ( z, q 2, µ 2 R) = Ce 2Φ I (â s,q 2,µ 2,z) ( Z I ( â s, µ 2 R, µ 2)) 2 ˆF I ( â s, Q 2, µ 2) 2 δ(1 z) â s renormalization and Z I makes partonic CS UV finite. Soft divergences from real radiation and virtual correction cancel upon summing over all the final states (KLN). Final state collinear singularities also cancel upon summing over all the final states (KLN) Still XSection is NOT fully finite due to initial state collinear singularities This can arise from virtual as well as real corrections Mass factorization to make collinear finite. Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 17

30 Cancellation of Divergences ˆ SV,I ( z, q 2, µ 2 R) = Ce 2Φ I (â s,q 2,µ 2,z) ( Z I ( â s, µ 2 R, µ 2)) 2 ˆF I ( â s, Q 2, µ 2) 2 δ(1 z) â s renormalization and Z I makes partonic CS UV finite. Soft divergences from real radiation and virtual correction cancel upon summing over all the final states (KLN). Final state collinear singularities also cancel upon summing over all the final states (KLN) Still XSection is NOT fully finite due to initial state collinear singularities This can arise from virtual as well as real corrections Mass factorization to make collinear finite. Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 17

31 Cancellation of Divergences ˆ SV,I ( z, q 2, µ 2 R) = Ce 2Φ I (â s,q 2,µ 2,z) ( Z I ( â s, µ 2 R, µ 2)) 2 ˆF I ( â s, Q 2, µ 2) 2 δ(1 z) â s renormalization and Z I makes partonic CS UV finite. Soft divergences from real radiation and virtual correction cancel upon summing over all the final states (KLN). Final state collinear singularities also cancel upon summing over all the final states (KLN) Still XSection is NOT fully finite due to initial state collinear singularities This can arise from virtual as well as real corrections Mass factorization to make collinear finite. Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 17

32 Factorization of Collinear Singularities ˆ SV,I ( z, q 2, µ 2 ) R = Γ T (z, µ 2 F ) SV,I ( z, q 2, µ 2 R, µ 2 ) F Γ(z, µ 2 F ) UV, Soft & Collinear finite Kernels Γ absorb all the initial state collinear singularities. Only diagonal elements contribute to threshold limit. Substituting this we arrive at the master formula to compute SV CS. Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 18

33 Factorization of Collinear Singularities ˆ SV,I ( z, q 2, µ 2 ) R = Γ T (z, µ 2 F ) SV,I ( z, q 2, µ 2 R, µ 2 ) F Γ(z, µ 2 F ) UV, Soft & Collinear finite Kernels Γ absorb all the initial state collinear singularities. Only diagonal elements contribute to threshold limit. Substituting this we arrive at the master formula to compute SV CS. Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 18

34 Master Formula for SV CS SV,I ( z, q 2, µ 2 R, µ 2 F ) = C exp ( Ψ I ( z, q 2, µ 2 R, µ 2 F, ɛ )) ɛ=0 where ( [ 2 Ψ I = ln Z I (â s, µ 2 R, µ 2, ɛ)] + ln ˆF I (â s, Q 2, µ 2, ɛ) 2) δ(1 z) + 2Φ I ( â s, q 2, µ 2, z, ɛ ) 2C ln Γ II (âs, µ 2, µ 2 F, z, ɛ ) Ravindran Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 19

35 Ingredients for SV CS 1. Logarithm of the overall operator renormalization ln Z I 2. Logarithm of the form factor ln ˆF I 3. Soft distribution Φ I 4. Logarithm of the mass factorization kernel ln Γ II Goal boils down to compute these quantities. Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 20

36 Ingredients for SV CS 1. Logarithm of the overall operator renormalization ln Z I 2. Logarithm of the form factor ln ˆF I 3. Soft distribution Φ I 4. Logarithm of the mass factorization kernel ln Γ II Goal boils down to compute these quantities. Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 20

37 Ingredient 1 : Overall Operator Renormalization ln Z I It satisfies the RGE d µ 2 R dµ 2 R ln Z I (â s, µ 2 R, µ 2, ɛ) = γi I s are anomalous dimensions. a i s(µ 2 R)γi 1 I i=1 Up to γ I 3 are available for I = q, b, g Solution of this RGE provides ln Z I. OOR is not required for DY Z I = 1. Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 21

38 Ingredient 2 : Form factor ln ˆF I ln ˆF I 2 can be computed from available results up to 3-loop. Instead, we have followed an alternative way. The reason would be clear soon. Gauge and RG invariance implies KG eqn Q 2 d dq 2 ln F ˆI = 1 ( ) ) [K I â s, µ2 R 2 µ 2, ɛ + G (â ] I s, Q2 µ 2, µ2 R R µ 2, ɛ Ashoke Sen, Mueller, Collins, Magnea K I contains all the poles in ɛ where, ST dimensions d = 4 + ɛ. G I collects finite terms at ɛ 0. Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 22

39 Ingredient 2... RG invariance of ˆF implies ) µ 2 d R K (â I s, µ2 R µ 2, ɛ = µ 2 R dµ 2 R d dµ 2 R ( ) G I â s, Q2, µ2 R µ 2, ɛ = A I ( a s (µ 2 ) R µ 2 R A I s are finite, called cusp anomalous dimensions. A I s are maximally non-abelian C A = N, C F = (N 2 1)/2N A g = C A C F A q Solve by expanding K I, G I and A I in powers of a s ( µ 2 R ). Substitute these K I & G I in KG eqn to solve for ln ˆF I to order by order. Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 23

40 Ingredient 2... RG invariance of ˆF implies ) µ 2 d R K (â I s, µ2 R µ 2, ɛ = µ 2 R dµ 2 R d dµ 2 R ( ) G I â s, Q2, µ2 R µ 2, ɛ = A I ( a s (µ 2 ) R µ 2 R A I s are finite, called cusp anomalous dimensions. A I s are maximally non-abelian C A = N, C F = (N 2 1)/2N A g = C A C F A q Solve by expanding K I, G I and A I in powers of a s ( µ 2 R ). Substitute these K I & G I in KG eqn to solve for ln ˆF I to order by order. Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 23

41 Ingredient 2... Solution of ln ˆF I The solution up to 3 loops are: ( ) ln ˆF I (â s, Q 2, µ 2, ɛ) = â i Q 2 i ɛ 2 i s S i=1 µ 2 ɛ I,(i) ˆL (ɛ) F with ˆL I,(1) (ɛ) = 1 { } F ɛ 2 2A I } {G I1 ɛ (ɛ), Moch, Vogt, Vermaseren; Ravindran; Magnea ˆL I,(2) (ɛ) = 1 { } F ɛ 3 β 0 A I { ɛ 2 12 } AI2 β 0G I1 (ɛ) + 1 { } 1 ɛ 2 GI 2 (ɛ), ˆL I,(3) (ɛ) = 1 { F ɛ 4 8 } 9 β2 0 AI { 2 ɛ 3 9 β 1A I β 0A I } 3 β2 0 GI 1 (ɛ) + 1 { ɛ 2 29 AI3 13 β 1G I1 (ɛ) 43 } β 0G I2 (ɛ) + 1 { } 1 ɛ 3 GI 3 (ɛ) At every order, all the poles except 1 ɛ previous order. can be predicted from Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 24

42 Ingredient 2... Solution of ln ˆF I The solution up to 3 loops are: ( ) ln ˆF I (â s, Q 2, µ 2, ɛ) = â i Q 2 i ɛ 2 i s S i=1 µ 2 ɛ I,(i) ˆL (ɛ) F with ˆL I,(1) (ɛ) = 1 { } F ɛ 2 2A I } {G I1 ɛ (ɛ), Moch, Vogt, Vermaseren; Ravindran; Magnea ˆL I,(2) (ɛ) = 1 { } F ɛ 3 β 0 A I { ɛ 2 12 } AI2 β 0G I1 (ɛ) + 1 { } 1 ɛ 2 GI 2 (ɛ), ˆL I,(3) (ɛ) = 1 { F ɛ 4 8 } 9 β2 0 AI { 2 ɛ 3 9 β 1A I β 0A I } 3 β2 0 GI 1 (ɛ) + 1 { ɛ 2 29 AI3 13 β 1G I1 (ɛ) 43 } β 0G I2 (ɛ) + 1 { } 1 ɛ 3 GI 3 (ɛ) At every order, all the poles except 1 ɛ previous order. can be predicted from Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 24

43 Ingredient 2... Single Pole Also Predictable 2-loop results for ˆF q & ˆF g in SU(N) dictates G I 1(ɛ) = 2 ( B I 1 γ I 1) + f I 1 + k=1 ɛ k g I,k 1 G I 2(ɛ) = 2 ( B2 I γ2 I ) + f I 2 2β 0 g I,1 1 + Origin : B I i collinear, γi i UV and f I i Ravindran, Smith, van Neerven k=1 ɛ k g I,k 2 soft region. B I i are collinear anomalous dimensions. Single pole can also be predicted! f I i are soft anomalous dimensions. These satisfy f g i = C A C F f q i i = 1, 2 Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 25

44 Ingredient 2... Single Pole Also Predictable 2-loop results for ˆF q & ˆF g in SU(N) dictates G I 1(ɛ) = 2 ( B I 1 γ I 1) + f I 1 + k=1 ɛ k g I,k 1 G I 2(ɛ) = 2 ( B2 I γ2 I ) + f I 2 2β 0 g I,1 1 + Origin : B I i collinear, γi i UV and f I i Ravindran, Smith, van Neerven k=1 ɛ k g I,k 2 soft region. B I i are collinear anomalous dimensions. Single pole can also be predicted! f I i are soft anomalous dimensions. These satisfy f g i = C A C F f q i i = 1, 2 Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 25

45 Ingredient 2... Single Pole Also Predictable 2-loop results for ˆF q & ˆF g in SU(N) dictates G I 1(ɛ) = 2 ( B I 1 γ I 1) + f I 1 + k=1 ɛ k g I,k 1 G I 2(ɛ) = 2 ( B2 I γ2 I ) + f I 2 2β 0 g I,1 1 + Origin : B I i collinear, γi i UV and f I i Ravindran, Smith, van Neerven k=1 ɛ k g I,k 2 soft region. B I i are collinear anomalous dimensions. Single pole can also be predicted! f I i are soft anomalous dimensions. These satisfy f g i = C A C F f q i i = 1, 2 Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 25

46 Ingredient 2... Single Pole Also Predictable 2-loop results for ˆF q & ˆF g in SU(N) dictates G I 1(ɛ) = 2 ( B I 1 γ I 1) + f I 1 + k=1 ɛ k g I,k 1 G I 2(ɛ) = 2 ( B2 I γ2 I ) + f I 2 2β 0 g I,1 1 + Origin : B I i collinear, γi i UV and f I i Ravindran, Smith, van Neerven k=1 ɛ k g I,k 2 soft region. B I i are collinear anomalous dimensions. Single pole can also be predicted! f I i are soft anomalous dimensions. These satisfy f g i = C A C F f q i i = 1, 2 Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 25

47 Ingredient 2... Single Pole Also Predictable 2-loop results for ˆF q & ˆF g in SU(N) dictates G I 1(ɛ) = 2 ( B I 1 γ I 1) + f I 1 + k=1 ɛ k g I,k 1 G I 2(ɛ) = 2 ( B2 I γ2 I ) + f I 2 2β 0 g I,1 1 + Origin : B I i collinear, γi i UV and f I i Ravindran, Smith, van Neerven k=1 ɛ k g I,k 2 soft region. B I i are collinear anomalous dimensions. Single pole can also be predicted! f I i are soft anomalous dimensions. These satisfy f g i = C A C F f q i i = 1, 2 Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 25

48 Ingredient 2... Single Pole Also Predictable 2-loop results for ˆF q & ˆF g in SU(N) dictates G I 1(ɛ) = 2 ( B I 1 γ I 1) + f I 1 + k=1 ɛ k g I,k 1 G I 2(ɛ) = 2 ( B2 I γ2 I ) + f I 2 2β 0 g I,1 1 + Origin : B I i collinear, γi i UV and f I i Ravindran, Smith, van Neerven k=1 ɛ k g I,k 2 soft region. B I i are collinear anomalous dimensions. Single pole can also be predicted! f I i are soft anomalous dimensions. These satisfy f g i = C A C F f q i i = 1, 2 Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 25

49 Ingredient 2... Single Pole Also Predictable 3-loop results for the ˆF I by Moch et al, Gehrmann et al. confirm this conclusion ( ) G I 3(ɛ) = 2(B3 I 3δ I,g β 2 ) + f3 I 2β 1 g I,1 1 2β 0 g I, β 0 g I,2 1 + ɛ k g I,k 3 with k=1 f I s are universal. f g 3 = C A C F f q 3 Explicit computation of FF gives ɛ dependent g I,k i. This understanding is crucial to get soft part. Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 26

50 Ingredient 2... Single Pole Also Predictable 3-loop results for the ˆF I by Moch et al, Gehrmann et al. confirm this conclusion ( ) G I 3(ɛ) = 2(B3 I 3δ I,g β 2 ) + f3 I 2β 1 g I,1 1 2β 0 g I, β 0 g I,2 1 + ɛ k g I,k 3 with k=1 f I s are universal. f g 3 = C A C F f q 3 Explicit computation of FF gives ɛ dependent g I,k i. This understanding is crucial to get soft part. Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 26

51 Ingredient 3 : Mass Factorization ln Γ DGLAP kernel Γ satisfies the RGE d µ 2 F dµ 2 F Γ(z, µ 2 F, ɛ) = 1 2 P ( z, µ 2 ) ( F Γ z, µ 2 F, ɛ ) P s are Altarelli Parisi splitting functions (matrix valued). Diagonal parts of the kernel ONLY contribute to SV limit. By solving we get Γ ln Γ Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 27

52 Ingredient 4 : Strategy to Calculate Soft Φ I Revisiting cancellation of poles ln Z I ln Γ II ln ˆF I Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 28

53 Ingredient 4 : Strategy to Calculate Soft Φ I Revisiting cancellation of poles ln Z I ln Γ II ln ˆF I Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 29

54 Ingredient 4 : Strategy to Calculate Soft Φ I Revisiting cancellation of poles ln Z I ln Γ II ln ˆF I Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 30

55 Ingredient 4 : Strategy to Calculate Soft Φ I Revisiting cancellation of poles ln Z I ln Γ II ln ˆF I Φ I Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 31

56 Ingredient 4 : Strategy to Calculate Soft Φ I Revisiting cancellation of poles ln Z I ln Γ II ln ˆF I Φ I Demanding finiteness of partonic CS Φ I must have soft as well as collinear contributions. Now onward we will call Φ I as Soft-Collinear distribution. Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 32

57 Ingredient 4 : Strategy to Calculate Soft Φ I Revisiting cancellation of poles ln Z I ln Γ II ln ˆF I Φ I Demanding finiteness of partonic CS Φ I must have soft as well as collinear contributions. Now onward we will call Φ I as Soft-Collinear distribution. Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 32

58 Ingredient 4... Ansatz for Soft-Collinear Φ I Demand of 1. finiteness of SV,I (z) at ɛ 0 and 2. the RGE d µ 2 R dµ 2 R Φ I ( â s, q 2, µ 2, z, ɛ ) = 0 can be achieved if we make an ansatz that Φ I satisfies KG type DE q 2 d dq 2 ΦI = 1 ( ) ) [K I â s, µ2 R 2 µ 2, z, ɛ + G (â ] I s, q2 µ 2, µ2 R R µ 2, z, ɛ K I contains all the poles in ɛ. G I contains finite terms in ɛ 0. Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 33

59 Ingredient 4... Ansatz for Soft-Collinear Φ I Demand of 1. finiteness of SV,I (z) at ɛ 0 and 2. the RGE d µ 2 R dµ 2 R Φ I ( â s, q 2, µ 2, z, ɛ ) = 0 can be achieved if we make an ansatz that Φ I satisfies KG type DE q 2 d dq 2 ΦI = 1 ( ) ) [K I â s, µ2 R 2 µ 2, z, ɛ + G (â ] I s, q2 µ 2, µ2 R R µ 2, z, ɛ K I contains all the poles in ɛ. G I contains finite terms in ɛ 0. Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 33

60 Ingredient 4... Ansatz for Soft-Collinear Φ I Demand of 1. finiteness of SV (z) at ɛ 0 and 2. the RGE d µ 2 R dµ 2 R Φ I ( â s, q 2, µ 2, z, ɛ ) = 0 can be achieved if we make an ansatz that Φ satisfies KG type DE q 2 d dq 2 ΦI = 1 ( ) ) [K I â s, µ2 R 2 µ 2, z, ɛ + G (â ] I s, q2 µ 2, µ2 R R µ 2, z, ɛ K I contains all the poles in ɛ. G I contains finite terms in ɛ 0. The solutions of KG consistent with our demands Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 34

61 Ingredient 4... Solution of Φ I Φ I (â s, q 2, µ 2, z, ɛ) = where This implies ˆφ I,(i) (ɛ) = i=1 â i s I,(i) ˆL F (ɛ) G I 1 (ɛ) = f I 1 + k=1 ( q 2 (1 z) 2 ( ɛ k G I,k 1 G I 2 (ɛ) = f I 2 2β 0G I,1 1 + µ 2 ) i ɛ ( ) 2 S i iɛ ɛ ˆφ I,(i) (ɛ) 1 z A I A I, G I (ɛ) G I (ɛ) ɛ k G I,k 2 k=1 ( ) G I 3 (ɛ) = f 3 I 2β 1G I,1 1 2β 0 G I, β 0 G I,2 1 + ɛ k G I,k 3 k=1 Explicit computation is required to get ɛ dependent cof ḠI,k i. ) Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 35

62 Ingredient 4... Solution of Φ I Φ I (â s, q 2, µ 2, z, ɛ) = where This implies ˆφ I,(i) (ɛ) = i=1 â i s I,(i) ˆL F (ɛ) G I 1 (ɛ) = f I 1 + k=1 ( q 2 (1 z) 2 ( ɛ k G I,k 1 G I 2 (ɛ) = f I 2 2β 0G I,1 1 + µ 2 ) i ɛ ( ) 2 S i iɛ ɛ ˆφ I,(i) (ɛ) 1 z A I A I, G I (ɛ) G I (ɛ) ɛ k G I,k 2 k=1 ( ) G I 3 (ɛ) = f 3 I 2β 1G I,1 1 2β 0 G I, β 0 G I,2 1 + ɛ k G I,k 3 k=1 Explicit computation is required to get ɛ dependent cof ḠI,k i. ) Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 35

63 Ingredient 4... Crucial Property of Φ I Explicit computation shows up to O(a 2 s). Φ q = C F Φ g and Ḡ q,k i = C F Ḡ g,k i C A C A soft-collinear distributions are universal Φ I = C I Φ with C I = { C A C F for Higgs in gg for DY and for Higgs in b b Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 36

64 Ingredient 4... Crucial Property of Φ I Explicit computation shows up to O(a 2 s). Φ q = C F Φ g and Ḡ q,k i = C F Ḡ g,k i C A C A soft-collinear distributions are universal Φ I = C I Φ with C I = { C A C F for Higgs in gg for DY and for Higgs in b b Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 36

65 Results : SV CS for DY at N 3 LO SV,(3) = SV,(3) δ δ(1 z) + 5 j=0 SV,(3) Dj D j All the D j, j = 0, 1,..., 5 and partial δ(1 z) contributions were known for a decade. We completed the full computation of δ(1 z) part. Moch and Vogt TA, Mahakhud, Rana, Ravindran All the required quantities were available except Ḡq,1 3 from Φ q. Recently, complete SV CS at N 3 LO for the Higgs production in gluon fusion was computed by Anastasiou et. al. From which we extracted Φ g at O(a 3 s). Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 37

66 Results : SV CS for DY at N 3 LO SV,(3) = SV,(3) δ δ(1 z) + 5 j=0 SV,(3) Dj D j All the D j, j = 0, 1,..., 5 and partial δ(1 z) contributions were known for a decade. We completed the full computation of δ(1 z) part. Moch and Vogt TA, Mahakhud, Rana, Ravindran All the required quantities were available except Ḡq,1 3 from Φ q. Recently, complete SV CS at N 3 LO for the Higgs production in gluon fusion was computed by Anastasiou et. al. From which we extracted Φ g at O(a 3 s). Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 37

67 Results : SV CS for DY at N 3 LO SV,(3) = SV,(3) δ δ(1 z) + 5 j=0 SV,(3) Dj D j All the D j, j = 0, 1,..., 5 and partial δ(1 z) contributions were known for a decade. We completed the full computation of δ(1 z) part. Moch and Vogt TA, Mahakhud, Rana, Ravindran All the required quantities were available except Ḡq,1 3 from Φ q. Recently, complete SV CS at N 3 LO for the Higgs production in gluon fusion was computed by Anastasiou et. al. From which we extracted Φ g at O(a 3 s). Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 37

68 Results : SV CS for DY at N 3 LO SV,(3) = SV,(3) δ δ(1 z) + 5 j=0 SV,(3) Dj D j All the D j, j = 0, 1,..., 5 and partial δ(1 z) contributions were known for a decade. We completed the full computation of δ(1 z) part. Moch and Vogt TA, Mahakhud, Rana, Ravindran All the required quantities were available except Ḡq,1 3 from Φ q. Recently, complete SV CS at N 3 LO for the Higgs production in gluon fusion was computed by Anastasiou et. al. From which we extracted Φ g at O(a 3 s). Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 37

69 Results : SV CS for DY at N 3 LO SV,(3) = SV,(3) δ δ(1 z) + 5 j=0 SV,(3) Dj D j All the D j, j = 0, 1,..., 5 and partial δ(1 z) contributions were known for a decade. We completed the full computation of δ(1 z) part. Moch and Vogt TA, Mahakhud, Rana, Ravindran All the required quantities were available except Ḡq,1 3 from Φ q. Recently, complete SV CS at N 3 LO for the Higgs production in gluon fusion was computed by Anastasiou et. al. From which we extracted Φ g at O(a 3 s). Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 37

70 Results : SV CS for DY at N 3 LO... Recall Φ q = C F C A Φ g was established up to O(a 2 s) by explicit computation. We conjectured the relation to be hold true even at O(a 3 s). Using this we got Φ q Ḡq,1 3. This gives the ONLY missing part to achieve δ(1 z) part. In addition, now D 7,... D 1 are available at N 4 LO exactly. TA, Mahakhud, Rana, Ravindran; de Florian, Mazzitelli, Moch, Vogt Later the result was reconfirmed by two groups : Catani, Cieri, de Florian, Ferrera, Frazzini; Li, von Manteuffel, Schabinger, Zhu This in turn establishes our conjecture at O(a 3 s). Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 38

71 Results : Numerical Implications At 14 TeV LHC with µ F = µ R = Q = m l + l [ Q 2 dσ dq ]δ [ 2 j Q 2 dσ Most of the contributions dq ]D 2 j from D j s are cancelled against δ(1 z) making N 3 LO SV more subleading. Q (GeV) δ (nb) D (nb) Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 39

72 Results : Numerical Implications... Correction at N 3 LO SV is very very small Good newz! Q (GeV) NNLO (nb) N 3 LO SV (nb) Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 40

73 Results : Numerical Implications... Scale dependence reduces NLO NNLO (i) R Q = 20 GeV 3 N LO SV Q = 200 GeV (i) R µ /Q R [ R (i) ] Q 2 d σ (i) (µ 2 dq 2 R ) / [ ] Q 2 d σ (i) (Q 2 ) and µ dq 2 F = Q Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 41

74 Summary A systematic way of computing threshold corrections to inclusive Higgs and DY productions in pqcd is prescribed. Underlying principles : factorization of soft and collinear divergences, RG invariance, gauge invariance and Sudakov resummation. δ(1 z) part of N 3 LO SV DY Xsection is computed for the first time. We find that the impact of the δ(1 z) contribution is quite large to the pure N 3 LO SV correction. This method has been later employed to compute some other inclusive and exclusive observables at threshold by us in Phys. Rev. Lett. 113 (2014) JHEP 1410 (2014) 139 JHEP 1502 (2015) 131 Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 42

75 Other Areas of Interest RG Improved Higgs Boson Production to N 3 LO in QCD TA, Das, Kumar, Rana, Ravindran arxiv: Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 43

76 Large Scale Uncertainties State of the art : The production cross section of Higgs boson through gg at N 3 LO by Anastasiou et. al. Consequence (a) Spectacular accuracy (b) Significant reduction in unphysical renormalization & factorization scales. Points of concern (a) Significant increase in scale uncertainties upon increasing the range of scale variation : µ R < m H /4 (b) Total cross section may become negative in this region. Reason : The presence of large logarithms of the scales at every order. Remedy : (a) Go beyond N 3 LO, (b) Perform all order resummations Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 44

77 Large Scale Uncertainties State of the art : The production cross section of Higgs boson through gg at N 3 LO by Anastasiou et. al. Consequence (a) Spectacular accuracy (b) Significant reduction in unphysical renormalization & factorization scales. Points of concern (a) Significant increase in scale uncertainties upon increasing the range of scale variation : µ R < m H /4 (b) Total cross section may become negative in this region. Reason : The presence of large logarithms of the scales at every order. Remedy : (a) Go beyond N 3 LO, (b) Perform all order resummations Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 44

78 Large Scale Uncertainties State of the art : The production cross section of Higgs boson through gg at N 3 LO by Anastasiou et. al. Consequence (a) Spectacular accuracy (b) Significant reduction in unphysical renormalization & factorization scales. Points of concern (a) Significant increase in scale uncertainties upon increasing the range of scale variation : µ R < m H /4 (b) Total cross section may become negative in this region. Reason : The presence of large logarithms of the scales at every order. Remedy : (a) Go beyond N 3 LO, (b) Perform all order resummations Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 44

79 Large Scale Uncertainties State of the art : The production cross section of Higgs boson through gg at N 3 LO by Anastasiou et. al. Consequence (a) Spectacular accuracy (b) Significant reduction in unphysical renormalization & factorization scales. Points of concern (a) Significant increase in scale uncertainties upon increasing the range of scale variation : µ R < m H /4 (b) Total cross section may become negative in this region. Reason : The presence of large logarithms of the scales at every order. Remedy : (a) Go beyond N 3 LO, (b) Perform all order resummations Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 44

80 Large Scale Uncertainties State of the art : The production cross section of Higgs boson through gg at N 3 LO by Anastasiou et. al. Consequence (a) Spectacular accuracy (b) Significant reduction in unphysical renormalization & factorization scales. Points of concern (a) Significant increase in scale uncertainties upon increasing the range of scale variation : µ R < m H /4 (b) Total cross section may become negative in this region. Reason : The presence of large logarithms of the scales at every order. Remedy : (a) Go beyond N 3 LO, (b) Perform all order resummations Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 44

81 Our Proposal to Improve µ R Dependence Our proposal : Perform resummations by including RG accessible µ R dependent logarithms of all orders. Consequence : Quite remarkable! µ = m F H MSTW TeV LHC LO NNLO NLO 3 N LO FO RESUM 50 [pb] σ µ / m R H Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 45

82 Our Proposal to Improve µ R Dependence FO NNLO RESUM NNLO 3 FO N LO 3 RESUM N LO [ pb ] σ µ F µ R = m H = [m H /10,10m H ] MSTW s [ TeV ] Figure: Dependence of µ R in both the fixed order and resummed Xsections on S Demerit : Method fails to improve the central value of Xsection, unlike other resummation. Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 46

83 Other Areas of Interest Two-loop QCD corrections to Higgs b + b + g amplitude TA, Mahakhud, Mathews, Rana, Ravindran JHEP 1408 (2014) 075 Two-Loop QCD Correction to massive spin-2 resonance 3 gluons TA, Mahakhud, Mathews, Rana, Ravindran JHEP 1405 (2014) 107 Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 47

84 Contact : taushif@imsc.res.in, ahmedtaushif@gmail.com Threshold Corrections To DY and Higgs at N 3 LO QCD INFN Sezione Di Torino 48